統(tǒng)計(jì)學(xué)方法與應(yīng)用課件lecture3

統(tǒng)計(jì)學(xué)方法及其統(tǒng)計(jì)學(xué)方法及其應(yīng)StatisticalMethodswithMinistryofEducationKeyLaboratoryofBioinformaticsBioinformaticsDivision,TNLIST/DepartmentofAutomationTsinghuaUniversity,Beijing100084,ChinaRuiJiang,PhDAssociateTheneedformultiplerandomTheneedformultiplerandom概描變數(shù)字特BivariateRandom統(tǒng)計(jì)學(xué)基BivariateRandom統(tǒng)計(jì)學(xué)基隨機(jī)變量的函“ArandomvariableisaquantitywhosevaluesarerandomandtowhichaprobabilitydistributionTossingtwofairTossingtwofairTossingtwofairdice,thesamplespaceistheCartesianproductoftwosets{1,2,3,4,5,6}=(1,1),(1,2),(1,3),(1,4),(1,5),(1,(2,1),(2,2),(2,3),(2,4),(2,5),(2,(3,1),(3,2),(3,3),(3,4),(3,5),(3,(4,1),(4,2),(4,3),(4,4),(4,5),(4,(5,1),(5,2),(5,3),(5,4),(5,5),(5,(6,1),(6,2),(6,3),(6,4),(6,5),(6,}UnivariaterandomX:RP(X=2)=1/?P(X=6)=5/?P(X=12)=1/X=UnivariaterandomX:RP(X=2)=1/?P(X=6)=5/?P(X=12)=1/X=Second d678956789456789345678923456781234567123456First X? X?HX(6,6)=UnivariaterandomY:RP(Y=0)=6/?P(Y=2)=8/?P(Y=5)=2/Y=UnivariaterandomY:RP(Y=0)=6/?P(Y=2)=8/?P(Y=5)=2/Y=|First-Second d654321054321014321012321012321012341012345123456FirstH(1,1)=Y?H(3,3)=Y?HY(6,6)=Bivariaterandom(X,Y): RHXY(1,1)=(2,0),?,HXY(3,3)=(6,0),?,HXY(6,6)=(12,Y=|First-Second d6543210543210143210123210123Bivariaterandom(X,Y): RHXY(1,1)=(2,0),?,HXY(3,3)=(6,0),?,HXY(6,6)=(12,Y=|First-Second d654321054321014321012321012321012341012345123456FirstXSecond d678956789456789345678923456781234567123456FirstBivariaterandomBivariaterandomBivariaterandomBivariaterandomAbivariaterandomvectorisafunctionfromasamplespaceS to?2,the2-dimensionalEuclideanspace.IntheoriginalIntheoriginalsamplespaceS(domainoftherandomvector),aprobabilityfunctioncanbedefined,e.g.P({(1,1)})=1/P({(1,4),(4,1)})=2/InthespaceR2(rangeoftherandomvector),aprobabilityfunctioncanbeinduced,e.g.,P(X=P(X=Y=0)=1/andY=3)=2/Probability23456789500000000004000000000300000000200000001000000000000Probability23456789500000000004000000000300000000200000001000000000000JointprobabilitymassJointLet(X,Y)beadiscretebivariaterandomvector.Thentheinto?definedf(x,y)=P(X=xf(x,y)from=yiscalledthejointprobabilitymassfunctionJointprobabilitymassJointLet(X,Y)beadiscretebivariaterandomvector.Thentheinto?definedf(x,y)=P(X=xf(x,y)from=yiscalledthejointprobabilitymassfunctionorjointpmfof(X,YdenotedbyfX,Y(x,y)foremphasizingtherandomvector(X,Y23456789500000000004000000000300000000200000001000000000000Marginal23456789Y500000000004000000000300000000200000001000000000000X23456789500000000004000000000300000000200000001000000000000Marginal23456789Y500000000004000000000300000000200000001000000000000X23456789500000000004000000000300000000200000001000000000000MarginalprobabilitymassMarginalLet(X,Y)beMarginalprobabilitymassMarginalLet(X,Y)beadiscretebivariaterandomvectorwith=x(x,y).ThenthemarginalpmfofX,fX(x)=PfXfX(x)=?fX,Y(x,yandthemarginalpmfofY,fY(y)=P=y),?f(y)f(x,YXxConditional2345678950000000000400000000030000000020000000100000000000023456789500000000004000000000300000000200000001000000000000XConditional2345678950000000000400000000030000000020000000100000000000023456789500000000004000000000300000000200000001000000000000XConditional2345678950000000000400000000030000000020000000100000000000023456789Y500000000004000000000300000000200000001000000000000Conditional2345678950000000000400000000030000000020000000100000000000023456789Y500000000004000000000300000000200000001000000000000ConditionalprobabilitymassConditionalLet(XConditionalprobabilitymassConditionalLet(X,Y)beadiscretebivariaterandomvactorwithjointf(x,y)andmarginalpmfsfX(x)fY(y).ForanyxsuchthatP(X=x)=fX(x)>X=xisthetheconditionalpmfofgiventhatof denotedbyf(y|x)anddefinedfX(xf(y|x)==y| =x)JointprobabilitydensityJointinto?iscalledJointprobabilitydensityJointinto?iscalledaAfunctionf(x,y)fromprobabilitydensityfunctionorjointpdfoftheì?,randomvector(X,Y),if,foreverydxd￥￥òòf(x,y)dxdy=- -MarginalprobabilitydensityMarginalLet(X,Y)beMarginalprobabilitydensityMarginalLet(X,Y)beacontinuousbivariaterandomvectorfX(xfX(x,y).ThenthemarginalpdfsofandYfY(y),aregiven￥òfX(x)fX-￥ConditionalprobabilitydensityConditionalLet(X,YConditionalprobabilitydensityConditionalLet(X,Y)beajointpdff(x,anyxsuchthatcontinuousbivariaterandomvactorandmarginalpdfsfX(x)fY(y).fX(x)>0,theconditionalpdfofthatX=xisthefunctionofy denotedbyf(y|x)anddefinedbyf(y|x)fX(xJointcumulativedistributionJointThejointcdfofthecontinuousistheJointcumulativedistributionJointThejointcdfofthecontinuousisthefuncitonF(x,y)defined(X,F(x,y)=P￡x￡forall(x,y)?RxyòF(x,y)- -=?2F(x,?xExpectationsofrandomIfg(X,Y)isExpectationsofrandomIfg(X,Y)isareal-valuedfunction,thenthevalueofg(X,Y)isdefinedto￥￥òg(x,y)f(x,Eg(X,Y)- -if(X,Y)isacontinuousrandomvector,?Eg(X,Y)g(x,y)f(x,y(x,yif(X,Y)isadiscreterandomConditionalConditionalIfg(Y)isareal-valuedfunctionofY,thentheexpectedConditionalConditionalIfg(Y)isareal-valuedfunctionofY,thentheexpectedvalueofg(Y)giventhatX=E?g(Y)|xùú?andisdefinedto￥òEg(Y)| g(y)f(y|??-inthecontinuouscasean?ù g(y)f(y| g(Y)|??yinthediscreteConditionalConditionalIfYaretwoindependentEConditionalConditionalIfYaretwoindependentEX=E(E(X|expectationsEX=òòxf(x,y=ò?êòxf(x|y)dx?f(y=òE(X|=E(E(X|YAvector(X,Y)issaidtoAvector(X,Y)issaidtohaveabivariatestandarddistributionifthepdfof(X,Y)-2rxy+y2x 2p1-expf(x,y|r)ú2(1-r2??2wherex,y?(-￥,￥),r?(-1,Now,whataremarginaldistributionsXandgivenY=ywhatistheconditionaldistributionofMarginalIf(X,Yhasabivariatestandardnormaldistribution,-2rxy+y2x expf(x,y)ú2(1-1-??￥=-￥MarginalIf(X,Yhasabivariatestandardnormaldistribution,-2rxy+y2x expf(x,y)ú2(1-1-??￥=-￥f(x,é-2rxy+y2- ￥ú=ò-￥expú2(1-1-??-2rxy+(rx)2+x2-(rx)2ùúy ￥exp=ò-￥2(1-1-?é (y-rx)2x2é? ￥-==÷?ê2)2(1-1-??x2 è2XNConditionalIf(X,Y)hasabivariatestandardnormal-2rxy+y2ConditionalIf(X,Y)hasabivariatestandardnormal-2rxy+y2x 2p1-expf(x,y)ú2(1-?? expy2f(y)è-2f(x,yf(x| (x-ry)2= exp-2(1-r2)ú 1-N(ry,1-??X|-2rxy+y2x expf(x,y -2rxy+y2x expf(x,y ú2(1-2p1-??2= exp?-x2; 2è?= exp?-y2. 2è?Ifr1thenf(x,y)=f(x)fthenf(x,y)1f(x)f(yInsomecases,thejointdistributionisequaltotheproductofmarginaldistributions;insomeothercases,theyarenotequal.Whenarethey= exp?-x2?;2(x-= exp?-x2?;2(x-ry)2 2p1-exp-2(1-r2)f(x|y)???2x1Ifr=Ifr1thenf(x|y)=-è?p22thenf(x|y)1fInsomecases,theconditionaldistributionisequaltothemarginaldistribution;insomeothercases,theyarenotequal.WhenaretheyLet(X,Y)beLet(X,Y)beaf(x,y)andmarginalrandomvectorwithjointpdfororpmfsfX(x)andfY(y).X arecalledindependentrandomvariablesif,foreveryx?andy?f(x,y)=XYXYareindependent,thepdfofgivenf(x,XYareindependent,thepdfofgivenf(x,yf(y|x)=fX(xfX(x)fYfX(x=fY(yis,theconditionalpdfisthesameasthemarginalthevalueofx.TheknowledgeofX=regardlessgivesnoadditionalinformationSufficientandLet(XSufficientandLet(X,Y)beabivariaterandomvectorwithjointpdforf(x,y).Thenand areindependentrandom andonlyifthereexistfunctionsg(x)andh(y)suchthat,everyx??andy?f(x,y)=g(xBecausef(x,y)iswe￥￥￥òf(x)=g(x)-￥h(f(x,y)dyg(xBecausef(x,y)iswe￥￥￥òf(x)=g(x)-￥h(f(x,y)dyg(xX--￥￥￥fY(y)=-￥f(x,y)dx=ò-￥g(x=h(y)-ù￥￥ò=êú(y)￥g(x)dxf(x)fg(xh??XY-￥￥=g(x=f(x,yò-￥ -￥h(y)dy?￥￥òò- -￥￥òòf(x,y=f(x,y- -=f(x,LetAìandbeindependentrandomvariables.ForLetAìandbeindependentrandomvariables.ForandBìP(X??)=P(X?A)P?events{?A}and{Y?B}are=òA=òAP(X??BfX,Y(x,yfX(x)fY==P(X?A)P(Y?BLetandbeindependentrandomvariables.Letg(X)befunctiononlyofandh(Y)Letandbeindependentrandomvariables.Letg(X)befunctiononlyofandh(Y)beafunctiononlyofy.ThenvariablesU=g(X)V=h(X)areForanyu?and ?R,={x:g(x)￡u}andBv={x:g(x)￡￡u,V￡vFU,V(u,v)=?Au,Y?Bv=P=P(X?Au)P(Y?Bv==Ff(u,(u,vUUùéù v?uLetand beindependentrandomvariables,letbeafunctiononlyofandh(y)Letand beindependentrandomvariables,letbeafunctiononlyofandh(y)beaonlyofy?g(X)??. òg(x)h(y)f(x,E[g(X)h(Y)]- - òjyg(x(x=- -=êùù￥￥ò(x)dxújh(yg(x--=E[g(X)]E[h(YMomentgeneratingFortwoindependentrandomXletMomentgeneratingFortwoindependentrandomXletg(x)=andh(y)=ety,E[g(X)]=E[etX]=E[h(Y)]=E[etY]=XYE[g(X)h(Y)]=E[et(X+Y)]=BecauseoftheE[g(X)h(Y)]=E[g(X)]E[h(Y)WethusXMX(t)=MX(t)MY(tSummationoftworandomLetandSummationoftworandomLetandbetwoindependentrandomvariablesgeneratingMX(t)MY(t).ThengeneratingfunctionoftherandomZ=XMZ(t)=MX(t)MY(tisSummationoftwoN(m,s2)andN(n,t2)Letrandomvariables, (tSummationoftwoN(m,s2)andN(n,t2)Letrandomvariables, (t)=exp÷t sm2èX2??1(t)=÷Mnt 2t2Y2(t)M(t)=exp?(m+n)t1(s2+t2)t2M(t)=ZXY2Z=XN(m+n,s2+t2UnivariatetransformationsofLetbearandomvariablewith.ForUnivariatetransformationsofLetbearandomvariablewith.ForY=g(X),thesampelspace={y:=g(x),?IfyfY(y)=P,=y?==P(X=xfX(xxg1(y?xg1(yIfy,fY(y)=BivariatetransformationsofLet(X,Y)beaBivariatetransformationsofLet(X,Y)beabivariaterandomvectorwith(U,V)beabivariateand =h(X,YajointfXy).definedbyU=g(X,Y={(x,y):fX,Y(x,y)>={(u,v):u=g(x,={(x,y):g(x,y)=andv=h(x,y)forsome(x,y)andh(x,y)=vforany(u,v)?fU,V(u,v)fX(x,yArandomvariableXissaidtohaveArandomvariableXissaidtohaveaPoisson(ldistributione-lP(X=x|),x=0,1,xwherelistheintensityTheprobabilityofanumberofeventsoccurringinafixedperiodoftimeiftheseeventsoccurwithaknownaveragerate(intensity)andindependentlyofthetimesincethelastevent==VarpmfandpmfandPoissondistributioninPoissondistributioninQuantileRandomSummationoftwoindependentLet andbetwoindependentPoissonrandomqandlrespectively.Thejointpmfof(X,Y)isSummationoftwoindependentLet andbetwoindependentPoissonrandomqandlrespectively.Thejointpmfof(X,Y)ise-e-l(x,y);y=fXxyU=+Y,VDefine={(x,y):x=;y=0,1, ={(u,v):v=0,1,;u=v+0,v+1,v+2,={(x,y):y=v= ;x=u-y=u-e-qqu-e-fU,V(u,v)=(u-v,v=,u=0,1,;u=v,v+1,vve-e-qqu-u?f(u)(u-Uvv=e-lu-v)!v!qu-(uue-(ql=(q+l,uX+Poisson(q+uUnivariatetransformationsofLethavepdffX(x),letY=gxUnivariatetransformationsofLethavepdffX(x),letY=gx),whereisa={x:fX(x)>function.SupposefX(x)iscontinuousonandg-1(y)hasacontinuousderivativeon={y:y=g(x)forsome?Thenthepdfofis--??f yf(y)XY?BivariatetransformationsofLet(X,Y)acontinuousbivariaterandomvectorwithjointBivariatetransformationsofLet(X,Y)acontinuousbivariaterandomvectorwithjoint(U,V)beabivariaterandomvectordefinedfX(x,y).U=g(X,Y)and =h(X,Y).={(x,y):fX,Y(x,y)>0}={(u,v):u=g(x,y)andv=h(x,y)forsome(x,y)Iftransformationisaone-to-transformation,fU,V(u,v)=fX(U,V),y(U,V))|X=jU,VY=yU,V ??u?y?u==-J.SummationoftworandomSummationoftworandomConvolutionLetthetransformationbeW=X andZ=X+Y.DefineX=jW,Z) and = ,Z)=ZWThentheJacobianis1,andthejointpdfis (w,z)=f(w)f(z-W ThemarginalpdffZ(z)isgiven fZ(z)=￥fW,Z(w,z)dw=￥fX(w)fY(z-Let(X,Y)betwoindependentcontinuousrandomvariableswithpdfsfX(x)andfY(y),thenthepdfofZ=X+Y ￥fZ(z)=￥fX(w)fY(z-Summationoftwoò￥f(w)Summationoftwoò￥f(w)f(z-w)dw=ò￥ exp?(w-m)2?1 exp?(z-w-n)2- - dw ?exp 2ps2+ +t) s2+ ê1s2+t2 t+(z-n)s2?2 ò- ex- ÷÷ú s2+ ? 1LetX N(m,s2)andY N(n,t2)betwoindependentnormalrandomvariables,thenZ=X N(m+n,s2+t2Z=X- N(m-n,s2+t2GammaGamma￥òG)a(0G(a+GammaGamma￥òG)a(0G(a+aa=>0==___p)=(n-GammafunctioninG(a+n)/G(a)=a(a (a+nG(a+1)/G( =G(a+2)/G(a =aG(a+n)/G(a+n-1)=a+n-Gammaf(x|shape=a,Gammaf(x|shape=a,scale=q)1￡x<,a>0,qEX=VarX=Gamma=-x,0￡x<￥,aGamma=-x,0￡x<￥,a>0,b>f(x|shape=,rate=xeaEX=aVar pdfandpdfandGammadistributioninQuantileRandomGammadistributioninQuantileRandomGamma(shape=1,Gamma1f(x|shape=Gamma(shape=1,Gamma1f(x|shape=a,scale=q)￡x<,a>0,qGamma(shape=1,scale=l-1f(x|)=lel,0￡x<￥,l>Gamma1f(x|shape=a,scale=q)￡x<,a>0,q￥ Gamma1f(x|shape=a,scale=q)￡x<,a>0,q￥ 1òM(t) 1x/e 0￥ò=1-x(1/t 0 ￥ò=xa-1eq/(1-qt0ùéù11úú=úq/(1-qtaa???=? è1-qtGammaM(t)=? è1-qtdM(t)=aq(1-qt)-1GammaM(t)=? è1-qtdM(t)=aq(1-qt)-1=m=dM(t)=a(a+1)q2(1-qt)-dx￠ =(a+=q22dM(t)=a(a+1)(a+2)q3(1-qt)-dx2￠ =(a+1)(a+=qb33sadM(t)=a(a+1)(a+2)(a+4)q4(1-qt)-dx6a￠ =(a+a+a+b-3q44kSummationoftwoGamma(a,scale=Gamma(b,scale=LetandSummationoftwoGamma(a,scale=Gamma(b,scale=Letandrandom? M(t)è1-qtX? ?bM(t)è1-qtY?1MZ(t)=MX(t)MY(t)=è1-qtZ=XGamma(a+b,scale=BetaLet Gamma(a,scale=q)and Gamma(b,scale=q)independentGammarandomvariables,ConsiderX=UBetaLet Gamma(a,scale=q)and Gamma(b,scale=q)independentGammarandomvariables,ConsiderX=U/(U+V),Y=U U=XY, =Y(1-XTheJacobianisyx=-=y(1-x)+xy=J1-é ùp(u,v) ua-1e-u/qvb-1e-v/qêéy(1é ùé ù(xy)a-1e-xy/qúb-e-y(1-x)/qúp(x,y)=ê???y(a+b)-1e-y/ùéG(a+=1ê(1-xa-b-x?Gamma(a+b,scale= Beta(a,BetaArandomvariableBetaArandomvariableissaidtohaveaBeta(a,b)distributionifthepdff(x|a,b)=G(a+b)xa-1(1-x)b-1,0<x<1,a>0,b>BetadistributioninQuantileBetadistributioninQuantileRandomG(a+b)xa-1(1-x)b-G(a+b)xa-1(1-x)b-isapdf,weG(a+b)xa-1(1-x)b-1ò=0InotheréG(a+b)ùG(a+1òxa-1(1-x)b-=ê=?G(a)G(b)0ThisiscalledabetaB(a,b)=ò1x-(1-x)-dx0G(a+CovarianceandCorrelationThecorrelationCovarianceandCorrelationThecorrelationof and isthenumberdefined =Cov(X,Y)X s ThecovarianceofX andY isthenumberdefinedbyrandomvariablesY-mCov(X,Y randomvariablesY-mCov(X,Y ?-mX+mmúY =EXY--mYEX+mX=EXY-mXmY-mYmX+mX=EXY-mXIfand areindependentrandom-mm=mm-mCov(X,Y)=EXY= =rIf andareanyrandomvariablesandIf andareanyrandomvariablesandand areanytwoconstants,)?Var(aX+bY)=EaX=E?aX-E(aX)+(bY-E(bY))?2=E?a(X-EX)+b(Y-EY)?2a?-EY)2?ù+Eé2ab(X-EX-EYEY=a2E(X-EX)2+b2E(Y-EY)2+2abE(X-EX=a2VarX+b2VarY+2abCov(X,YIf andareindependentrandomvariables,Var(aX+bY)=a2VarX+b2VarYLinearLinearConsidertheLinearLinearConsidertheh(t =E((X-m)t+(Y-m) =s2t2+2Cov(X,Y)t+ andits(2Cov(X,Y))2-4s2s2￡ Foranyrandom X -1￡rX ￡ =1ifandonlyifthereexistnumbersa10andXsuchthatP(Y=aX+b)=1.If =1,thena>Xandif =-1,thena<XBivariatenormalArandomvector(X,Y)issaidtohasdistributioniftheirjointpdf1,r)′f(x,y|m,BivariatenormalArandomvector(X,Y)issaidtohasdistributioniftheirjointpdf1,r)′f(x,y|m,,22 1-2ps ???x-m??y-m??2ú yexp?2-2rY÷+XXY÷2(1-r2) ?? ssXXYYBivariatenormal x=(x,y??rs÷,S=X Bivariatenormal x=(x,y??rs÷,S=X è÷ -rs=?1Y )rss21-2 X Bivariatenormaldistributionp(x|m,S)= exp-(2p)|SBivariatenormalIngeneral,ZBivariatenormalIngeneral,Z=aX+b hasaN(m,s2)distribution, m=am+bm =a +2abrs +b2s2 W=X- hasaN(m,s2)distribution, =m-m =s2-2rs +s2 If(X,Y bivariatenormal(m,m,s2,s2,r),the distributionofXisN(m,s2),themarginaldistribution isN(m,s2 Correlation(X)ù÷?Y--cov(X,Y==rsssX XY??￥?X-÷?￥òòf(x,Correlation(X)ù÷?Y--cov(X,Y==rsssX XY??￥?X-÷?￥òòf(x,yY=- -￥sXsYMakethechangeof?? ÷?Y=?X-X-?s÷tY=X ÷?X s??èYs,s+s t òr=st+)sfXXXYt- -Notice|trXt2 -2rs+/t)2=(s-rt2) +(1-r2)tttst2?ù??(s-rt2 ￥￥òè-21 ?2p(1-r2)t2ES=rtt2t￥=r-= - BivariatenormalBivariatenormalNoNegativeHierarchicalHierarchicalTheHierarchicalHierarchicalThedistributionofarandomvariabledependsonaquantitythatalsohasadistribution.Beta-Binomial(n,q),Beta(a,b).Letnp(x|èx?=G(a+b)qa-1(1Beta-Binomial(n,q),Beta(a,b).Letnp(x|èx?=G(a+b)qa-1(1-q)b-?Gn÷òx(1-bn-x-qò1p(x)=p(q)p(x|q)dqG(a)G(b)èxù1?n=éG(a+b)G(a+b+núèxú G(a+x)G(b+n-x??AccordingBaye's p(q)p(xp(xq)n ==p(q|xa+x(1-bnxq11G(a+x)G(b+n-x~Beta(a+x,b+n-xq|LaplaceIf Exponentiall),thatisf(x)=1e-x/lx30,l>0,LaplaceIf Exponentiall),thatisf(x)=1e-x/lx30,l>0,l1ey/lY=-Xyieldsf(y)f(x)Inotherl1e-x/l1e-y/f(y)(x)ll(-Now,thehierarchical1e-z/f(z|b=1)f(z|b=0)l1ez/(-lBWef(z)=f(z|b=1)P(b=1)+f(z|b=0)P(b1l1l==(z)ez/(z(-1e-|z|/lThispdfdefinesaLaplacedistribution,akadoubleexponentialdistributionLaplaceversusnormal(equalLaplaceversusnormal(equalXYLaplaceNormalMultivariateRandom統(tǒng)計(jì)學(xué)基MultivariateRandom統(tǒng)計(jì)學(xué)基隨機(jī)變量的函“ArandomvariableisaquantitywhosevaluesarerandomandtowhichaprobabilitydistributionMultivariaterandomRandomXMultivariaterandomRandomX={X1,X2,…, ={x1,x2,…,Ann-dimensionalrandomvectorisafunctionfromasamplespaceS to?n,then-dimensionalEuclideanspace.MultivariaterandomS→S→S→MultivariaterandomS→S→S→pmformarginal,andconditionalpmformarginal,andconditionalpmforDerivative(conditionalexpectations,Mutualindependence,Pair-wisecorrelationJointDiscreteContinuousf(JointDiscreteContinuousf(x)=f ,xn F(x)= ,xn)=ò- f(x ,xn)dx dxA ,xn)dx dxf(x)=f ,xn)=P(X1=x ,X =xF(x)=F ,xn) f ,tt1￡x1,,tnP(X?A)=?fMarginalDiscreteContinuous MarginalDiscreteContinuous xkf f ,xk,xk ,xn)dxk dxf ,xk) f ,xk,xk ,xn(xk+1 ,xn)??n-ConditionalConditional)=f ,xkConditionalConditional)=f ,xk,xk,xnf,,|x,kn1kf(x ,x1kYi=gi(X)=gi(XYi=gi(X)=gi(X,XnXi =hi(Y)=hi,YnJfX(x)=f ,xnfY(y)=f,hn)|JEf(x,,xEf(x,,x ,,x)g(x1 ,xk) kk k ,xk f ,xk|xk ,x dx dx- - f(x1,,xn-￥ò-￥f(x1,,xk,xk+1,,xk)dx1dxEf(x,,x ,xk) - - ,xk ,xk dx dx -￥ò-￥f(x1,,xk,xk+1,,xk)dxk+1dxEg(X)=?g(x)f( Eg(X)= ò-￥g(x)f(x)MutuallyindependentrandomMutuallyMutuallyindependentrandomMutuallyMutuallyindependent1pairwiseLetX1,?,Xn berandomvariableswithjointpdforpmff(x1,?,xn).LetfX(xi)denotethemarginalpdfiorpmfofXi.ThenX1,?,Xn arecalledmutuallyindependentrandomvariablesif,forevery(x1,?,xn),nf(x1,?,xn) fX(xiiiLetX1,?,LetX1,?,Xn bemutuallyindependentrandomvariableswithmgfMX(t),?,MX(t).LetZ=X1+ +Xn.Then themgfof nM(t) i Inparticular,ifX,?, allhavethesame withmgfMX(t),nMZ(t)=[MX(t)]LetX1,?,Xnbemutuallyindependentrandomvariables.Letg1,?,gnbereal-valuedfunctionssuchthatgi(xi)isafunctiononyofxi,i=1,,n.ThenE E[g(x) i i i SummationofmultipleGamma(a,scale=q),i=1,?,n,inSummationofmultipleGamma(a,scale=q),i=1,?,n,inLetXGammarandomvariables, (t)=? ÷è1+XLetZ=X1MZ(t)?ni?an i(t)=M÷è1-qtii??niniZa,scale=~Gamma(XiiBernoulliABernoullitrialisanexperimentBernoulliABernoullitriali